Letters in Mathematical Physics

, Volume 106, Issue 7, pp 913–923

Incompatibility of Time-Dependent Bogoliubov–de-Gennes and Ginzburg–Landau Equations

  • Rupert L. Frank
  • Christian Hainzl
  • Benjamin Schlein
  • Robert Seiringer
Open Access


We study the time-dependent Bogoliubov–de-Gennes equations for generic translation-invariant fermionic many-body systems. For initial states that are close to thermal equilibrium states at temperatures near the critical temperature, we show that the magnitude of the order parameter stays approximately constant in time and, in particular, does not follow a time-dependent Ginzburg–Landau equation, which is often employed as a phenomenological description and predicts a decay of the order parameter in time. The full non-linear structure of the equations is necessary to understand this behavior.


superconductivity quasi-free states critical temperature BCS theory 

Mathematics Subject Classification

82D50 46N50 

Copyright information

© The Author(s) 2016

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Rupert L. Frank
    • 1
  • Christian Hainzl
    • 2
  • Benjamin Schlein
    • 3
  • Robert Seiringer
    • 4
  1. 1.Mathematics 253-37CaltechPasadenaUSA
  2. 2.Mathematisches InstitutUniversität TübingenTübingenGermany
  3. 3.Institute of MathematicsUniversity of ZürichZurichSwitzerland
  4. 4.Institute of Science and Technology Austria (IST Austria)KlosterneuburgAustria

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